报告题目 (Title)：Low regularity ill-posedness for 3D elastic waves and for 3D ideal compressible MHD driven by shock formation (激波驱动的三维弹性波方程和可压MHD方程低正则性的不适定性)
报告人 (Speaker)：Xinliang An (National University of Singapore)
报告时间 (Time)：2021年12月15日(周三)10:00
报告地点 (Place)：腾讯会议 会议ID 628-301-795 密码2795
邀请人(Inviter)：刘见礼
报告摘要：We construct counterexamples to the local existence of low-regularity solutions to elastic wave equations and to the ideal compressible magnetohydrodynamics (MHD) system in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad’s classic results on the scalar wave equation by showing that the Cauchy problems for 3D elastic waves and for 3D MHD system are ill-posed in $H^3(R^3)$ and $H^2(R^3)$, respectively. Both elastic waves and MHD are physical systems with multiple wave-speeds. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. In particular, when the magnetic field is absent in MHD, we also provide a desired low-regularity ill-posedness result for the 3D compressible Euler equations, and it is sharp with respect to the regularity of the fluid velocity. Our proofs for elastic waves and for MHD are based on a coalition of a carefully designed algebraic approach and a geometric approach. To trace the nonlinear interactions of various waves, we algebraically decompose the 3D elastic waves and the 3D ideal MHD equations into $6\times 6$ and $7\times 7$ non-strictly hyperbolic systems. Via detailed calculations, we reveal their hidden subtle structures. With them we give a complete description of solutions’dynamics up to the earliest singular event, when a shock forms. This talk is based on joint works with Haoyang Chen and Silu Yin.